Optimal. Leaf size=167 \[ -\frac {(1+a)^2 (1-a-b x)^{5/2}}{b^3 \sqrt {1+a+b x}}-\frac {\left (11+18 a+6 a^2\right ) \sqrt {1-a-b x} \sqrt {1+a+b x}}{2 b^3}-\frac {\left (11+18 a+6 a^2\right ) (1-a-b x)^{3/2} \sqrt {1+a+b x}}{6 b^3}-\frac {(1-a-b x)^{5/2} \sqrt {1+a+b x}}{3 b^3}-\frac {\left (11+18 a+6 a^2\right ) \text {ArcSin}(a+b x)}{2 b^3} \]
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Rubi [A]
time = 0.14, antiderivative size = 167, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 7, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6298, 91, 81,
52, 55, 633, 222} \begin {gather*} -\frac {\left (6 a^2+18 a+11\right ) \text {ArcSin}(a+b x)}{2 b^3}-\frac {\left (6 a^2+18 a+11\right ) \sqrt {a+b x+1} (-a-b x+1)^{3/2}}{6 b^3}-\frac {\left (6 a^2+18 a+11\right ) \sqrt {a+b x+1} \sqrt {-a-b x+1}}{2 b^3}-\frac {\sqrt {a+b x+1} (-a-b x+1)^{5/2}}{3 b^3}-\frac {(a+1)^2 (-a-b x+1)^{5/2}}{b^3 \sqrt {a+b x+1}} \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 55
Rule 81
Rule 91
Rule 222
Rule 633
Rule 6298
Rubi steps
\begin {align*} \int e^{-3 \tanh ^{-1}(a+b x)} x^2 \, dx &=\int \frac {x^2 (1-a-b x)^{3/2}}{(1+a+b x)^{3/2}} \, dx\\ &=-\frac {(1+a)^2 (1-a-b x)^{5/2}}{b^3 \sqrt {1+a+b x}}+\frac {\int \frac {(1-a-b x)^{3/2} \left (-(1+a) (3+2 a) b+b^2 x\right )}{\sqrt {1+a+b x}} \, dx}{b^3}\\ &=-\frac {(1+a)^2 (1-a-b x)^{5/2}}{b^3 \sqrt {1+a+b x}}-\frac {(1-a-b x)^{5/2} \sqrt {1+a+b x}}{3 b^3}-\frac {\left (11+18 a+6 a^2\right ) \int \frac {(1-a-b x)^{3/2}}{\sqrt {1+a+b x}} \, dx}{3 b^2}\\ &=-\frac {(1+a)^2 (1-a-b x)^{5/2}}{b^3 \sqrt {1+a+b x}}-\frac {\left (11+18 a+6 a^2\right ) (1-a-b x)^{3/2} \sqrt {1+a+b x}}{6 b^3}-\frac {(1-a-b x)^{5/2} \sqrt {1+a+b x}}{3 b^3}-\frac {\left (11+18 a+6 a^2\right ) \int \frac {\sqrt {1-a-b x}}{\sqrt {1+a+b x}} \, dx}{2 b^2}\\ &=-\frac {(1+a)^2 (1-a-b x)^{5/2}}{b^3 \sqrt {1+a+b x}}-\frac {\left (11+18 a+6 a^2\right ) \sqrt {1-a-b x} \sqrt {1+a+b x}}{2 b^3}-\frac {\left (11+18 a+6 a^2\right ) (1-a-b x)^{3/2} \sqrt {1+a+b x}}{6 b^3}-\frac {(1-a-b x)^{5/2} \sqrt {1+a+b x}}{3 b^3}-\frac {\left (11+18 a+6 a^2\right ) \int \frac {1}{\sqrt {1-a-b x} \sqrt {1+a+b x}} \, dx}{2 b^2}\\ &=-\frac {(1+a)^2 (1-a-b x)^{5/2}}{b^3 \sqrt {1+a+b x}}-\frac {\left (11+18 a+6 a^2\right ) \sqrt {1-a-b x} \sqrt {1+a+b x}}{2 b^3}-\frac {\left (11+18 a+6 a^2\right ) (1-a-b x)^{3/2} \sqrt {1+a+b x}}{6 b^3}-\frac {(1-a-b x)^{5/2} \sqrt {1+a+b x}}{3 b^3}-\frac {\left (11+18 a+6 a^2\right ) \int \frac {1}{\sqrt {(1-a) (1+a)-2 a b x-b^2 x^2}} \, dx}{2 b^2}\\ &=-\frac {(1+a)^2 (1-a-b x)^{5/2}}{b^3 \sqrt {1+a+b x}}-\frac {\left (11+18 a+6 a^2\right ) \sqrt {1-a-b x} \sqrt {1+a+b x}}{2 b^3}-\frac {\left (11+18 a+6 a^2\right ) (1-a-b x)^{3/2} \sqrt {1+a+b x}}{6 b^3}-\frac {(1-a-b x)^{5/2} \sqrt {1+a+b x}}{3 b^3}+\frac {\left (11+18 a+6 a^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{4 b^2}}} \, dx,x,-2 a b-2 b^2 x\right )}{4 b^4}\\ &=-\frac {(1+a)^2 (1-a-b x)^{5/2}}{b^3 \sqrt {1+a+b x}}-\frac {\left (11+18 a+6 a^2\right ) \sqrt {1-a-b x} \sqrt {1+a+b x}}{2 b^3}-\frac {\left (11+18 a+6 a^2\right ) (1-a-b x)^{3/2} \sqrt {1+a+b x}}{6 b^3}-\frac {(1-a-b x)^{5/2} \sqrt {1+a+b x}}{3 b^3}-\frac {\left (11+18 a+6 a^2\right ) \sin ^{-1}(a+b x)}{2 b^3}\\ \end {align*}
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Mathematica [A]
time = 0.12, size = 190, normalized size = 1.14 \begin {gather*} -\frac {\sqrt {-b} \left (-52+2 a^4+33 b x+26 b^2 x^2-9 b^3 x^3+2 b^4 x^4+a^3 (51+2 b x)+a^2 (50+69 b x)+a \left (-51+106 b x+9 b^2 x^2+2 b^3 x^3\right )\right )+6 \left (11+18 a+6 a^2\right ) \sqrt {b} \sqrt {1-a^2-2 a b x-b^2 x^2} \sinh ^{-1}\left (\frac {\sqrt {-b} \sqrt {1-a-b x}}{\sqrt {2} \sqrt {b}}\right )}{6 (-b)^{7/2} \sqrt {-((-1+a+b x) (1+a+b x))}} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(625\) vs.
\(2(143)=286\).
time = 0.08, size = 626, normalized size = 3.75
method | result | size |
risch | \(\frac {\left (2 b^{2} x^{2}-2 a b x +2 a^{2}-9 b x +27 a +28\right ) \left (b^{2} x^{2}+2 a b x +a^{2}-1\right )}{6 b^{3} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}-\frac {3 \arctan \left (\frac {\sqrt {b^{2}}\, \left (x +\frac {a}{b}\right )}{\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}\right ) a^{2}}{b^{2} \sqrt {b^{2}}}-\frac {9 \arctan \left (\frac {\sqrt {b^{2}}\, \left (x +\frac {a}{b}\right )}{\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}\right ) a}{b^{2} \sqrt {b^{2}}}-\frac {11 \arctan \left (\frac {\sqrt {b^{2}}\, \left (x +\frac {a}{b}\right )}{\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}\right )}{2 b^{2} \sqrt {b^{2}}}-\frac {4 \sqrt {-b^{2} \left (x +\frac {1+a}{b}\right )^{2}+2 b \left (x +\frac {1+a}{b}\right )}\, a^{2}}{b^{4} \left (x +\frac {1}{b}+\frac {a}{b}\right )}-\frac {8 \sqrt {-b^{2} \left (x +\frac {1+a}{b}\right )^{2}+2 b \left (x +\frac {1+a}{b}\right )}\, a}{b^{4} \left (x +\frac {1}{b}+\frac {a}{b}\right )}-\frac {4 \sqrt {-b^{2} \left (x +\frac {1+a}{b}\right )^{2}+2 b \left (x +\frac {1+a}{b}\right )}}{b^{4} \left (x +\frac {1}{b}+\frac {a}{b}\right )}\) | \(363\) |
default | \(\frac {2 \left (-1-a \right ) \left (\frac {\left (-b^{2} \left (x +\frac {1+a}{b}\right )^{2}+2 b \left (x +\frac {1+a}{b}\right )\right )^{\frac {5}{2}}}{b \left (x +\frac {1+a}{b}\right )^{2}}+3 b \left (\frac {\left (-b^{2} \left (x +\frac {1+a}{b}\right )^{2}+2 b \left (x +\frac {1+a}{b}\right )\right )^{\frac {3}{2}}}{3}+b \left (-\frac {\left (-2 b^{2} \left (x +\frac {1+a}{b}\right )+2 b \right ) \sqrt {-b^{2} \left (x +\frac {1+a}{b}\right )^{2}+2 b \left (x +\frac {1+a}{b}\right )}}{4 b^{2}}+\frac {\arctan \left (\frac {\sqrt {b^{2}}\, \left (x +\frac {1+a}{b}-\frac {1}{b}\right )}{\sqrt {-b^{2} \left (x +\frac {1+a}{b}\right )^{2}+2 b \left (x +\frac {1+a}{b}\right )}}\right )}{2 \sqrt {b^{2}}}\right )\right )\right )}{b^{4}}+\frac {\frac {\left (-b^{2} \left (x +\frac {1+a}{b}\right )^{2}+2 b \left (x +\frac {1+a}{b}\right )\right )^{\frac {3}{2}}}{3}+b \left (-\frac {\left (-2 b^{2} \left (x +\frac {1+a}{b}\right )+2 b \right ) \sqrt {-b^{2} \left (x +\frac {1+a}{b}\right )^{2}+2 b \left (x +\frac {1+a}{b}\right )}}{4 b^{2}}+\frac {\arctan \left (\frac {\sqrt {b^{2}}\, \left (x +\frac {1+a}{b}-\frac {1}{b}\right )}{\sqrt {-b^{2} \left (x +\frac {1+a}{b}\right )^{2}+2 b \left (x +\frac {1+a}{b}\right )}}\right )}{2 \sqrt {b^{2}}}\right )}{b^{3}}+\frac {\left (1+a \right )^{2} \left (-\frac {\left (-b^{2} \left (x +\frac {1+a}{b}\right )^{2}+2 b \left (x +\frac {1+a}{b}\right )\right )^{\frac {5}{2}}}{b \left (x +\frac {1+a}{b}\right )^{3}}-2 b \left (\frac {\left (-b^{2} \left (x +\frac {1+a}{b}\right )^{2}+2 b \left (x +\frac {1+a}{b}\right )\right )^{\frac {5}{2}}}{b \left (x +\frac {1+a}{b}\right )^{2}}+3 b \left (\frac {\left (-b^{2} \left (x +\frac {1+a}{b}\right )^{2}+2 b \left (x +\frac {1+a}{b}\right )\right )^{\frac {3}{2}}}{3}+b \left (-\frac {\left (-2 b^{2} \left (x +\frac {1+a}{b}\right )+2 b \right ) \sqrt {-b^{2} \left (x +\frac {1+a}{b}\right )^{2}+2 b \left (x +\frac {1+a}{b}\right )}}{4 b^{2}}+\frac {\arctan \left (\frac {\sqrt {b^{2}}\, \left (x +\frac {1+a}{b}-\frac {1}{b}\right )}{\sqrt {-b^{2} \left (x +\frac {1+a}{b}\right )^{2}+2 b \left (x +\frac {1+a}{b}\right )}}\right )}{2 \sqrt {b^{2}}}\right )\right )\right )\right )}{b^{5}}\) | \(626\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains complex when optimal does not.
time = 0.48, size = 622, normalized size = 3.72 \begin {gather*} \frac {{\left (-b^{2} x^{2} - 2 \, a b x - a^{2} + 1\right )}^{\frac {3}{2}} a^{2}}{b^{5} x^{2} + 2 \, a b^{4} x + a^{2} b^{3} + 2 \, b^{4} x + 2 \, a b^{3} + b^{3}} + \frac {2 \, {\left (-b^{2} x^{2} - 2 \, a b x - a^{2} + 1\right )}^{\frac {3}{2}} a}{b^{5} x^{2} + 2 \, a b^{4} x + a^{2} b^{3} + 2 \, b^{4} x + 2 \, a b^{3} + b^{3}} - \frac {{\left (-b^{2} x^{2} - 2 \, a b x - a^{2} + 1\right )}^{\frac {3}{2}} a}{b^{4} x + a b^{3} + b^{3}} - \frac {6 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} a^{2}}{b^{4} x + a b^{3} + b^{3}} + \frac {{\left (-b^{2} x^{2} - 2 \, a b x - a^{2} + 1\right )}^{\frac {3}{2}}}{b^{5} x^{2} + 2 \, a b^{4} x + a^{2} b^{3} + 2 \, b^{4} x + 2 \, a b^{3} + b^{3}} - \frac {{\left (-b^{2} x^{2} - 2 \, a b x - a^{2} + 1\right )}^{\frac {3}{2}}}{b^{4} x + a b^{3} + b^{3}} - \frac {12 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} a}{b^{4} x + a b^{3} + b^{3}} - \frac {3 \, a^{2} \arcsin \left (b x + a\right )}{b^{3}} - \frac {6 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{b^{4} x + a b^{3} + b^{3}} + \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 4 \, b x + 4 \, a + 3} x}{2 \, b^{2}} - \frac {9 \, a \arcsin \left (b x + a\right )}{b^{3}} + \frac {{\left (-b^{2} x^{2} - 2 \, a b x - a^{2} + 1\right )}^{\frac {3}{2}}}{3 \, b^{3}} + \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 4 \, b x + 4 \, a + 3} a}{2 \, b^{3}} - \frac {3 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} a}{b^{3}} - \frac {i \, \arcsin \left (b x + a + 2\right )}{2 \, b^{3}} - \frac {6 \, \arcsin \left (b x + a\right )}{b^{3}} + \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 4 \, b x + 4 \, a + 3}}{b^{3}} - \frac {3 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.42, size = 159, normalized size = 0.95 \begin {gather*} \frac {3 \, {\left (6 \, a^{3} + {\left (6 \, a^{2} + 18 \, a + 11\right )} b x + 24 \, a^{2} + 29 \, a + 11\right )} \arctan \left (\frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (b x + a\right )}}{b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right ) - {\left (2 \, b^{3} x^{3} - 7 \, b^{2} x^{2} + 2 \, a^{3} + {\left (16 \, a + 19\right )} b x + 53 \, a^{2} + 103 \, a + 52\right )} \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{6 \, {\left (b^{4} x + {\left (a + 1\right )} b^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2} \left (- \left (a + b x - 1\right ) \left (a + b x + 1\right )\right )^{\frac {3}{2}}}{\left (a + b x + 1\right )^{3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.45, size = 166, normalized size = 0.99 \begin {gather*} -\frac {1}{6} \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (x {\left (\frac {2 \, x}{b} - \frac {2 \, a b^{6} + 9 \, b^{6}}{b^{8}}\right )} + \frac {2 \, a^{2} b^{5} + 27 \, a b^{5} + 28 \, b^{5}}{b^{8}}\right )} + \frac {{\left (6 \, a^{2} + 18 \, a + 11\right )} \arcsin \left (-b x - a\right ) \mathrm {sgn}\left (b\right )}{2 \, b^{2} {\left | b \right |}} + \frac {8 \, {\left (a^{2} + 2 \, a + 1\right )}}{b^{2} {\left (\frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left | b \right |} + b}{b^{2} x + a b} + 1\right )} {\left | b \right |}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^2\,{\left (1-{\left (a+b\,x\right )}^2\right )}^{3/2}}{{\left (a+b\,x+1\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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